**LECTURER IN MATHEMATICS COLLEGIATE EDUCATION**

**(CATEGORY No. 53/2015)**

**MATHEMATICS**

**History of Development of Mathematics.**

**Module-I**

Mensuration, length of arcs, area of sectors of circles,
tangents to circles, circumcircle and incircle of polygons, area of polygons,
solids-volume and surface area.

Fundamentals of mumber theory. Continued fractions.

Boolean Algebra

Fundamentals of graph theory.

**Module -II**

Sets and binary operations, groups, Sylon's Theorems, Rings
and ideals, Fields, extension fields, rings of polynomials, finite fields,
Galois Theory, constructible numbers.

System of Linear Equations -Vector spaces, linear
transformations, characteristic values, characteristic polynomial, Minimal polynomial,
Cayley-Hamilton theorem, triangulation and diagonalization of metrics.

Hyperspaces and linear functionals.

**Module -III**

Normed spaces, Banach spaces and related theorems, Linear Maps, inner product spaces, Hilbert spaces
and related theorems, finite dimensional and infinite dimensional normed
spaces, bounded operators, spectrum, duals and transposes. Adjoints, normal,
unitary and self adjoint operators. Polynomial Equations, Trigonometry,
Analytical geometry of two dimension and three dimension, similarity of
triangles, vectors, matrices.

Calculus, applications of differentiation and integration,
elemetary functions (logarithms, exponential, hyperbolic, trigonometric etc),
Fundamental theorem of calculus, mean value theorems, maxima and
minima-functions of more than one
independent variables, derivatives, partial derivatives, saddle point, critical
point.

**Module -IV**

Real numbers, rational, irrational numbers, algebraic and
order properties of Real numbers, supremum property, countable and uncountable
sets, completeness property, sequences and series of red numbers, relations and
functions, limits and continuity of functions, uniform continuity, differentiability
and integrability of functions, Riemann integral, Riemann-Stieltges integral, sequences
and series of functions. Term by term differentiation and integration of series
of functions.

Lebesgue measure, lebesgue integral, convergence theorems
and applications

**Module -V**

Complex numbers, De Moirre's Theorem, Algebraic properties
of complex numbers, regions in the complex plane.

Complex functions, analytic functions, harmonic functions,
conformal mapping, elemetary functions, derivatives and integrals of complex
functions and related theorems, sigularities, residue theorem and its
applications, Power series, Taylor series, Laurent series.

Metric spaces, topological spaces, basis, subbasis, closed
set, closure, interior, boundary, neighbourhood. Connectedness and compactness,
locally connected, path connected, locally compact spaces.

Functions, continuous functions, homeomorphism, quotient
space.

Seperation axioms and related theorems.

**Module -VI**

First order ordinary differential equations-formation,
properties and various methods of solving.

Picards method of approximation.

Numerical methods

Second order ordinary differential equations – formaiton,
properties and various methods of solving.

Equidimensional equations.

Existence and uniqueness of solutions.

Systems of first order equations.

Series solutions of first order and second order ordinary
differential equation at ordinary adn regular singular points.

Hypergeometric functions and equations, legendre equations
and polynomials. Chebyshev's Equations and polynomials. Bessels equations and
Functions.

Laplace transform, fourier series, beta and Gamma functions.

Formation and solution of first order partial differential
equation in two independent variables. Functional dependence, analytic
functions. Second order partial differential equation, formation, classification.

Wave equation, heat diffusion equation, laplace equation.

Numerical solutions of algebraic equations, finite
differences, interpolation.

Module -VII

Fundamentals of Theory of Wavelets, Fuzzy set theory,
Fractal geometry, Modular functions Jordan forms, elliptic functions, Riemann
Zets Function, Automate and formal languages, Block Designs.

Monodromy theorem, Reimann mapping theorem, producttopology
and Tychnoff theorem.

Solutions at infinity of Differential Equations, Integral
Equations, calculus of Variations.

Fundamentals of differential geometry, contractions, inverse
function theorem, implicit function theorem.

Fundamentals of Mechanics and Fundamentals of FluidDynamics.

**Module VIII**

**RESEARCH METHODOLOGY/TEACHING APTITUDE**

**I. TEACHING APTITUDE**

- Teaching: Nature, objectives, characteristics and basic requirements;
- Learner's characteristics;
- Factors affecting teaching;
- Methods of teaching;
- Teaching aids;
- Evaluation systems.

**II. RESEARCH APTITUDE**

- Research: Meaning, Characteristics and types;
- Steps of research;
- Methods of research;
- Research Ethics;
- Paper, article, workshop, seminar, conference and symposium;
- Thesis writing: its characteristics and format.

**Module IX (a )**

Salient Features of Indian Constitution

Salient features of the Constitution - Preamble- Its
significance and its place in the interpretation of the Constitution.

Fundamental Rights - Directive Principles of State Policy - Relation between Fundamental

Rights and Directive Principles - Fundamental Duties.

Executive - Legislature - Judiciary - Both at Union and
State Level. - Other Constitutional Authorities.

Centre-State Relations - Legislative - Administrative and
Financial.

Services under the Union and the States.

Emergency Provisions.

Amendment Provisions of the Constitution.

**Module IX (b)**

Social Welfare Legislations and Programmes

Social Service Legislations like Right to Information Act,
Prevention of atrocities against Women & Children, Food Security Act,
Environmental Acts etc. and Social Welfare Programmes like Employment Guarantee
Programme, Organ and Blood Donation etc.

**Module X (a)**

**RENAISSANCE IN KERALA**

**TOWARDS A NEW SOCIETY**

Introduction to English education - various missionary
organisations and their functioning- founding of educational institutions,
factories, printing press etc.

**EFFORTS TO REFORM THE SOCIETY**

(A) Socio-Religious reform Movements

SNDP Yogam, Nair Service Society, Yogakshema Sabha, Sadhu
Jana Paripalana Sangham, Vaala Samudaya Parishkarani Sabha, Samathwa Samajam,
Islam Dharma Paripalana Sangham, Prathyaksha Raksha Daiva Sabha, Sahodara
Prasthanametc.

(B) Struggles and Social Revolts

Upper cloth revolts.Channar agitation, Vaikom Sathyagraha,
Guruvayoor Sathyagraha, Paliyam Sathyagraha. Kuttamkulam Sathyagraha, Temple
Entry Proclamation, Temple Entry Act .Malyalee Memorial, Ezhava Memorial etc.

Malabar riots, Civil Disobedience Movement, Abstention
movement etc.

**ROLE OF PRESS IN RENAISSANCE**

Malayalee, Swadeshabhimani, Vivekodayam, Mithavadi, Swaraj,
Malayala Manorama, Bhashaposhini, Mathnubhoomi, Kerala Kaumudi, Samadarsi,
Kesari, AI-Ameen, Prabhatham, Yukthivadi, etc

**AWAKENING THROUGH LITERATURE**

Novel, Drama, Poetry, Purogamana Sahithya Prasthanam, Nataka
Prashtanam, Library movement etc

**WOMEN AND SOCIAL CHANGE**

Parvathi Nenmenimangalam, Arya Pallam, A V Kuttimalu Amma,
Lalitha Prabhu.Akkamma Cheriyan, Anna Chandi, Lalithambika Antharjanam
andothers

**LEADERS OF RENAISSANCE**

Thycaud Ayya Vaikundar, Sree Narayana Guru, Ayyan
Kali.Chattampi Swamikal, Brahmananda Sivayogi, Vagbhadananda, Poikayil Yohannan(Kumara
Guru) Dr Palpu, Palakkunnath Abraham Malpan, Mampuram Thangal, Sahodaran
Ayyappan, Pandit K P Karuppan, Pampadi John Joseph, Mannathu Padmanabhan, V T
Bhattathirippad, Vakkom Abdul Khadar Maulavi, Makthi Thangal, Blessed Elias
Kuriakose Chaavra, Barrister G P Pillai, TK Madhavan, Moorkoth Kumaran, C. Krishnan,
K P Kesava Menon, Dr.Ayyathan Gopalan, C
V Kunjuraman, Kuroor Neelakantan Namboothiripad, Velukkutty Arayan, K P
Vellon, P K Chathan Master, K Kelappan,
P. Krishna Pillai, A K Gopalan, T R Krishnaswami Iyer, C Kesavan. Swami Ananda
Theerthan , M C Joseph, Kuttippuzha Krishnapillai and others

**LITERARY FIGURES**

Kodungallur Kunhikkuttan Thampuran, KeralaVarma Valiyakoyi
Thampuran, Kandathil Varghesc Mappila. Kumaran Asan, Vallathol Narayana Menon,
Ulloor S Parameswara Iyer, G Sankara Kurup, Changampuzha Krishna Pillai, Chandu
Menon, V aikom Muhammad Basheer. Kesav Dev, Thakazhi Sivasankara Pillai,
Ponkunnam Varky, S K Pottakkad and others

**Module X (b )**

**GENERAL KNOWLEDGE AND CURRENT AFFAIRS**

General Knowledge and Current Affairs

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